Finite Group Databasebeta

← #30612608

Subgroups — #30612608

Order 15616.

634 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21462633C211yesmaximal, characteristic, complemented
2#21462625C211yesmaximal, characteristic, complemented
3#21461432C211yesmaximal, characteristic, complemented
4#21461428C211yesmaximal, characteristic, complemented
5#21461425C211yesmaximal, characteristic, complemented
6#21461421C211yesmaximal, characteristic, complemented
7Q16×#92420C211yesmaximal, characteristic, complemented
8#21461403C211yesmaximal, characteristic, complemented
9SD16×#92420C211yesmaximal, characteristic, complemented
10#21461399C211yesmaximal, characteristic, complemented
11#21461391C211yesmaximal, characteristic, complemented
12#21461153C211yesmaximal, characteristic, complemented
13#21461131C211yesmaximal, characteristic, complemented
14#21461025C211yesmaximal, characteristic, complemented
15#21455836C211yesmaximal, characteristic
16#16840813C2211yescharacteristic
17Q8×#92420C2211yescharacteristic
18Q8×#92420C2211yescharacteristic
19Q8×#92420C2211yescharacteristic
20#16840797C2211yescharacteristic
21#16840794—21no—
22#16840790—21no—
23#16840790—21no—
24#16840790C2211yescharacteristic
25#16840790C2211yescharacteristic
26#16840789—21no—
27D4×#92420C2211yescharacteristic
28#16840786—21no—
29#16840784C2211yescharacteristic
30#16840781—21no—
31#16840781C2211yescharacteristic
32#16840781C2211yescharacteristic
33#16840726—21no—
34C4×D61⋊C8—21no—
35C4×D61⋊C8C2211yescharacteristic
36#16840626C2211yescharacteristic
37#16840625C2211yescharacteristic
38#16840624C2211yescharacteristic
39#16840623C2211yescharacteristic
40Q8⋊#92420C2211yescharacteristic
41Q8⋊#92420C2211yescharacteristic
42Q8⋊#92420C2211yescharacteristic
43Q8⋊#92420C2211yescharacteristic
44#16840610C2211yescharacteristic
45#16840610C2211yescharacteristic
46D4⋊#92420C2211yescharacteristic
47D4⋊#92420C2211yescharacteristic
48#92420⋊C8C2211yescharacteristic
49#92420⋊C8C2211yescharacteristic
50C8×#92420C2211yescharacteristic
51C8×#92420C2211yescharacteristic
52#16840291C2211yescharacteristic
53C4×#11758226C2211yescharacteristic
54#16840101C411yescharacteristic
55#16840084C411yescharacteristic
56#16840069C411yescharacteristic
57#16840063C411yescharacteristic
58#16839974C411yescharacteristic
59#16839923C411yescharacteristic
60#16839906C411yescharacteristic
61Q16×Dic61C2211yescharacteristic
62SD16×Dic61C2211yescharacteristic
63#16839847C2211yescharacteristic
64#16839842C2211yescharacteristic
65#16839814C2211yescharacteristic
66C61×#464C411yescharacteristic
67C2×#11758217—21no—
68C2×D61⋊C8—21no—
69C2×D61⋊C8—21no—
70C2×D61⋊C8D411yescharacteristic
71C2×D61⋊C8D411yescharacteristic
72C22×#92420—21no—
73#14519844—21no—
74C4⋊#92420—21no—
75C4⋊#92420—21no—
76C4⋊#92420—21no—
77C4⋊#92420—21no—
78C4⋊#92420C2311yescharacteristic
79C4⋊#92420C2311yescharacteristic
80C4⋊#92420C2311yescharacteristic
81C4⋊#92420C2311yescharacteristic
82#14519834—21no—
83#14519834—21no—
84#14519834—21no—
85#14519834—21no—
86#14519834—21no—
87C22⋊#92420—21no—
88C22⋊#92420—21no—
89#14519832—21no—
90#14519832—21no—
91C4×#92420—21no—
92C4×#92420—21no—
93C4×#92420—21no—
94C4×#92420—21no—
95C4×#92420D411yescharacteristic
96C4×#92420D411yescharacteristic
97C4×#92420C2311yescharacteristic
98C4×#92420C2311yescharacteristic
99C4×#92420C2311yescharacteristic
100C4×#92420C2311yescharacteristic
101C2×#11758226C2×C411yescharacteristic
102C2×#11758201C2×C411yescharacteristic
103C2×#11758199C2×C411yescharacteristic
104#14519770—21no—
105#14519769C2×C411yescharacteristic
106#14519759—21no—
107#14519753C2×C411yescharacteristic
108#14519748C2×C411yescharacteristic
109Q8×Dic61C2311yescharacteristic
110Q8×Dic61C2311yescharacteristic
111C4×D244—21no—
112C4×D244C2×C411yescharacteristic
113D4×Dic61C2311yescharacteristic
114#14519739—21no—
115#14519739C2311yescharacteristic
116#14519737—21no—
117#14519735—21no—
118#14519735C2×C411yescharacteristic
119#14519733—21no—
120C42×D61—21no—
121C42×D61C2311yescharacteristic
122#14519711C2×C411yescharacteristic
123#14519710C2×C411yescharacteristic
124#14519709C2×C411yescharacteristic
125#14519708C2×C411yescharacteristic
126C4⋊#92418C2×C411yescharacteristic
127#14519701C2×C411yescharacteristic
128#14519701C2×C411yescharacteristic
129#14519701C2×C411yescharacteristic
130#14519699C2×C411yescharacteristic
131#14519699C2×C411yescharacteristic
132#14519698C2×C411yescharacteristic
133#14519689C2×C411yescharacteristic
134#14519689C2×C411yescharacteristic
135C8×Dic61C2311yescharacteristic
136C8×Dic61C2311yescharacteristic
137C2×Q16×C61C2×C411yescharacteristic
138C2×SD16×C61C2×C411yescharacteristic
139#124×C61C2×C411yescharacteristic
140C4×Q8×C61C2×C411yescharacteristic
141C4⋊C8×C61C2×C411yescharacteristic
142#103×C61C2×C411yescharacteristic
143#102×C61C2×C411yescharacteristic
144C2×Q8×C61C22×C411yescharacteristic
145C2×Q8×C61C22×C411yescharacteristic
146C2×D4×C61C22×C411yescharacteristic
147C2×C22⋊D61—21no—
148C22×Dic61—21no—
149#11758226—22no—
150#11758226—22no—
151C2×D244—21no—
152C2×D244C22×C411yescharacteristic
153C2×C4×D61—21no—
154C2×C4×D61—21no—
155C2×C4×D61—21no—
156C2×C4×D61C22×C411yescharacteristic
157C2×C4×D61C2×D411yescharacteristic
158C2×C4×D61#4111yescharacteristic
159#11758217—42no—
160C2×#92420—21no—
161C2×#92420—21no—
162C2×#92420—21no—
163C2×#92420—21no—
164C2×#92420—21no—
165C2×#92420—21no—
166C2×#92420—42no—
167C2×#92420C2×D411yescharacteristic
168C2×#92420C2×D411yescharacteristic
169C2×#92420C2×D411yescharacteristic
170C2×#92420C2×D411yescharacteristic
171D61⋊C8—22no—
172D61⋊C8—22no—
173D61⋊C8—42no—
174D61⋊C8—42no—
175Q16×C61—22no—
176SD16×C61—22no—
177C2×C8×C61C22×C411yescharacteristic
178C2×C8×C61C22×C411yescharacteristic
179C4⋊C4×C61—21no—
180C4⋊C4×C61C22×C411yescharacteristic
181C22⋊C4×C61—21no—
182C22⋊C4×C61—21no—
183C42×C61—21no—
184C42×C61C22×C411yescharacteristic
185C22⋊Dic61—21no—
186C22⋊Dic61—21no—
187#11758201—22no—
188#11758199—22no—
189#11758198—21no—
190#11758198—21no—
191#11758198—21no—
192#11758198—21no—
193C4⋊Dic61—21no—
194C4⋊Dic61—21no—
195C4⋊Dic61C22×C411yescharacteristic
196C4⋊Dic61C22×C411yescharacteristic
197C4⋊Dic61C22×C411yescharacteristic
198#11758196—21no—
199#11758196—21no—
200#11758196—21no—
201C4×Dic61—21no—
202C4×Dic61—21no—
203C4×Dic61—21no—
204C4×Dic61C2×D411yescharacteristic
205C4×Dic61#4111yescharacteristic
206C4×Dic61C2411yescharacteristic
207C2×#92418C22×C411yescharacteristic
208C2×#92418C22×C411yescharacteristic
209C23×C61—21no—
210C22×D61—21no—
211C22×D61C4×D411yescharacteristic
212Q8×C61—21no—
213Q8×C61—21no—
214Q8×C61—21no—
215Q8×C61#13112yes—
216D4×C61—21no—
217D4×C61—21no—
218C2×C4×C61—21no—
219C2×C4×C61—21no—
220C2×C4×C61—21no—
221C2×C4×C61C4×D411yescharacteristic
222C2×C4×C61C4×D411yescharacteristic
223C2×C4×C61C23×C411yescharacteristic
224C22⋊D61—42no—
225C2×Dic61—21no—
226C2×Dic61—21no—
227C2×Dic61—21no—
228C2×Dic61—41no—
229C2×Dic61—41no—
230C2×Dic61C4×D411yescharacteristic
231C2×Dic61C22×D411yescharacteristic
232C2×Dic61C2×#4111yescharacteristic
233D244—24no—
234D244—21no—
235D244#13112yes—
236C4×D61—24no—
237C4×D61—21no—
238C4×D61—21no—
239C4×D61—21no—
240C4×D61—21no—
241C4×D61—41no—
242C4×D61—42no—
243C4×D61—41no—
244C4×D61—41no—
245#92420—21no—
246#92420—21no—
247#92420—41no—
248#92420—41no—
249#92420—41no—
250#92420—41no—
251#92420—41no—
252#92420—41no—
253#92420#13512yes—
254#92420#13512yes—
255C8×C61—21no—
256C8×C61—21no—
257#92418—21no—
258#92418—21no—
259#17904—611nomaximal, Sylow
260C22×C61—41no—
261C22×C61—41no—
262C22×C61—41no—
263C22×C61C2×C4×D411yescharacteristic
264C2×D61—21no—
265C2×D61—22no—
266C2×D61—21no—
267C2×D61—21no—
268C2×D61—41no—
269C2×D61—41no—
270C4×C61—22no—
271C4×C61—21no—
272C4×C61—21no—
273C4×C61—41no—
274C4×C61—41no—
275C4×C61C2×#13111yescharacteristic
276C4×C61C2×#13111yescharacteristic
277Dic61—21no—
278Dic61—21no—
279Dic61—41no—
280Dic61—41no—
281Dic61—41no—
282Dic61C2×#13511yescharacteristic
283Dic61C2×#13511yescharacteristic
284#3102—611no—
285#2966—611no—
286C8×#41—611no—
287#1584—611no—
288#1583—611no—
289#1582—611no—
290#1581—611no—
291C8×Q16—611no—
292C8×SD16—611no—
293C8×SD16—611no—
294C8×D8—611no—
295#1572—611no—
296#1488—611no—
297#1482—611no—
298#1450—611no—
299C2×C61—81no—
300C2×C61#292911yescharacteristic
301C2×C61C4×#13511yescharacteristic
302C2×C61#295511yescharacteristic
303D61—41no—
304D61—42no—
305#529—611no—
306C4×#41—611no—
307#485—611no—
308#482—611no—
309#479—611no—
310#478—611no—
311#464#692311yescharacteristic
312#463—611no—
313#452—611no—
314#449—611no—
315#446—611no—
316C8×Q8—611no—
317C8×Q8—611no—
318C8×Q8—611no—
319C4×Q16—611no—
320C4×SD16—611no—
321C4×SD16—611no—
322C4×D8—611no—
323#436—611no—
324#435—1221no—
325C8×D4—611no—
326C8×D4—611no—
327C8×D4—611no—
328C8×D4—1221no—
329C8×D4—1221no—
330C8×D4—1221no—
331#433—1221no—
332#432—611no—
333#431—611no—
334#431—611no—
335#431—611no—
336#431—1221no—
337#405—1221no—
338C2×C4×C8—611no—
339C2×C4×C8—1221no—
340#335—611no—
341#335—611no—
342#334—611no—
343#334—611no—
344Q8⋊C8—611no—
345Q8⋊C8—611no—
346Q8⋊C8—611no—
347Q8⋊C8—611no—
348D4⋊C8—611no—
349D4⋊C8—611no—
350D4⋊C8—611no—
351D4⋊C8—611no—
352#322—611no—
353#322—611no—
354C82—611no—
355C82—611no—
356C61#1790411yescharacteristic, complemented, Sylow
357C2×#41—611no—
358C2×Q16C2×#692311yescharacteristic
359C2×SD16—611no—
360C2×SD16C2×#692311yescharacteristic
361C2×D8—611no—
362C2×M16—1221no—
363C22×C8—611no—
364C22×C8—611no—
365C22×C8—1221no—
366C22×C8—1221no—
367C22×C8—1221no—
368#126—1221no—
369#125—611no—
370#124C2×#692311yescharacteristic
371#123—1221no—
372C22⋊Q8—611no—
373#121—611no—
374C4×Q8—611no—
375C4×Q8—611no—
376C4×Q8C2×#692311yescharacteristic
377C4×D4—611no—
378C4×D4—611no—
379C4×D4—611no—
380C4×D4—1221no—
381C4×D4—1221no—
382C4×D4—1221no—
383#117—611no—
384#117—1221no—
385#117—1221no—
386C2×C42—611no—
387C2×C42—1221no—
388#107—611no—
389#107—611no—
390#106—611no—
391#106—611no—
392C4⋊C8—611no—
393C4⋊C8—611no—
394C4⋊C8—611no—
395C4⋊C8—611no—
396C4⋊C8—611no—
397C4⋊C8—1221no—
398C4⋊C8—1221no—
399C4⋊C8—1221no—
400C4⋊C8—1221no—
401C4⋊C8—1221no—
402C4⋊C8C2×#692311yescharacteristic
403#103—611no—
404#103—611no—
405#103—611no—
406#103C2×#692311yescharacteristic
407#102—611no—
408#102—611no—
409#102—611no—
410#102C2×#692311yescharacteristic
411C22⋊C8—611no—
412C22⋊C8—611no—
413C22⋊C8—1221no—
414C22⋊C8—1221no—
415C22⋊C8—1221no—
416C22⋊C8—1221no—
417C22⋊C8—1221no—
418C22⋊C8—1221no—
419C22⋊C8—1221no—
420#97—1221no—
421#97—1221no—
422C4×C8—611no—
423C4×C8—611no—
424C4×C8—611no—
425C4×C8—611no—
426C4×C8—611no—
427C4×C8—611no—
428C4×C8—611no—
429C4×C8—611no—
430C4×C8—1221no—
431C4×C8—1221no—
432C4×C8—1221no—
433C4×C8—1221no—
434#41—1222no—
435#41—1222no—
436C2×Q8C22×#692311yescharacteristic
437C2×Q8C22×#692311yescharacteristic
438C2×D4—611no—
439C2×D4—1221no—
440C2×D4—1221no—
441C2×D4C22×#692311yescharacteristic
442C22×C4—611no—
443C22×C4—611no—
444C22×C4—611no—
445C22×C4—1221no—
446C22×C4—1221no—
447C22×C4—1221no—
448C22×C4—1221no—
449Q16—22no—
450SD16—22no—
451SD16—1222no—
452D8—1222no—
453M16—2442no—
454C2×C8—611no—
455C2×C8—611no—
456C2×C8—611no—
457C2×C8—611no—
458C2×C8—611no—
459C2×C8—611no—
460C2×C8—1222no—
461C2×C8—1222no—
462C2×C8—1221no—
463C2×C8—1221no—
464C2×C8—1221no—
465C2×C8—1221no—
466C2×C8—1221no—
467C2×C8—1221no—
468C2×C8—2442no—
469C2×C8—2442no—
470C2×C8—2442no—
471C2×C8C22×#692311yescharacteristic
472C2×C8C22×#692311yescharacteristic
473C4⋊C4—21no—
474C4⋊C4—611no—
475C4⋊C4—611no—
476C4⋊C4—611no—
477C4⋊C4—1221no—
478C4⋊C4—1221no—
479C4⋊C4—1221no—
480C4⋊C4—1221no—
481C4⋊C4—1221no—
482C4⋊C4C22×#692311yescharacteristic
483C22⋊C4—21no—
484C22⋊C4—21no—
485C22⋊C4—1221no—
486C22⋊C4—1221no—
487C22⋊C4—1221no—
488C22⋊C4—1221no—
489C22⋊C4—1221no—
490C22⋊C4—1221no—
491C42—21no—
492C42—611no—
493C42—611no—
494C42—611no—
495C42—1221no—
496C42—1221no—
497C42—1221no—
498C42C22×#692311yescharacteristic
499C23—21no—
500C23—611no—
501C23—1221no—
502Q8—21no—
503Q8—21no—
504Q8—21no—
505Q8#1451986412yes—
506D4—21no—
507D4—21no—
508D4—612no—
509D4—1224no—
510D4—1221no—
511D4—2442no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—611no—
516C2×C4—611no—
517C2×C4—611no—
518C2×C4—1221no—
519C2×C4—1221no—
520C2×C4—1224no—
521C2×C4—1221no—
522C2×C4—1221no—
523C2×C4—1221no—
524C2×C4—1221no—
525C2×C4—1221no—
526C2×C4—2441no—
527C2×C4—2442no—
528C2×C4—2441no—
529C2×C4—2441no—
530C2×C4—2441no—
531C2×C4—2441no—
532C2×C4D4×#692311yescharacteristic
533C2×C4D4×#692311yescharacteristic
534C2×C4C23×#692311yescharacteristic
535C8—21no—
536C8—21no—
537C8—612no—
538C8—612no—
539C8—1221no—
540C8—1221no—
541C8—1221no—
542C8—1221no—
543C8—2441no—
544C8—2441no—
545C8—2441no—
546C8—2441no—
547C8—2441no—
548C8—2441no—
549C22—41no—
550C22—41no—
551C22—41no—
552C22—1222no—
553C22—1221no—
554C22—1221no—
555C22—1221no—
556C22—2441no—
557C22—2441no—
558C22C2×D4×#692311yescharacteristic
559C4—22no—
560C4—21no—
561C4—21no—
562C4—41no—
563C4—41no—
564C4—611no—
565C4—611no—
566C4—1221no—
567C4—1221no—
568C4—2441no—
569C4—2441no—
570C4—2441no—
571C4C2×#1451986411yescharacteristic
572C4D61⋊#13111yescharacteristic
573C2—81no—
574C2—2441no—
575C2—2442no—
576C2#2146253311yescharacteristic
577C2#135×#692311yescharacteristic
578C2#2146259211yescharacteristic
subgroupG/H
centerC22C2×D4×#6923
commutatorC2×C4×C61C23×C4
FrattiniC2×C4C23×#6923
FittingC61×#464C4
socleC22×C61C2×C4×D4
prime radical#16839906C4
cosocleC4×Dic61C24
hypercenter#464#6923
hypocenterC61#17904
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #464

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #21462633C2semidirect
2. #21462625C2semidirect
3. #21461432C2semidirect
4. #21461428C2semidirect
5. #21461425C2semidirect
6. #21461421C2semidirect
7. Q16×#92420C2semidirect
8. #21461403C2semidirect
9. SD16×#92420C2semidirect
10. #21461399C2semidirect
11. #21461391C2semidirect
12. #21461153C2semidirect
13. #21461131C2semidirect
14. #21461025C2semidirect
356. C61#17904semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#214626332
#214626252
#214614322
#214614282
#214614252
#214614212
Q16×#924202
#214614032
SD16×#924202
#214613992
#214613912
#214611532
#214611312
#214610252
#214558362
#1790461